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# How the Law of Large Numbers Affects Forex Traders

The law of large numbers (LLN) is a theorem from mathematics, which states that as the number of trials for testing some probabilistic outcome increases, the average of the obtained results becomes closer to the expected value. In other words, the more measurements you have in your sample, the more confidence you have in your results. Although a rare Forex course does not mention the importance of learning the basics of math statistics and probabilities, they rarely focus on the LLN and its significance to practical trading.

## Examples

### A toss of a coin

A basic example of the law of large numbers is a sequence of coin tosses. When you toss up a coin and see how it lands, you either get heads or tails with equal probability of 50% each. However, if you toss it just one or few times, the average outcome might be quite different from the equal number of heads and tails. Nevertheless, thanks to the LLN, the more tosses your produce, the closer the average outcome becomes to 50%. On the chart below, you can see how the average value of the coin toss outcomes (where 0 is heads and 1 is tails) becomes closer to 0.5 as the number of experiments grows:

Another example comes directly from currency trading. Suppose, you have a profitable strategy with equal chance of winning and equal chance of losing (Pw = Pl = 0.5), but each losing trade is just \$3, while each wining trade is \$5. The expected profit per trade is 0.5 × \$5 - 0.5 × \$3 = \$1. The chart below demonstrates potential net profit from a series of 100 trades. It plots the total net profit (orange line) versus the expected total net profit (blue line):

As you can see, if you made only first 20 trades from this series, you would be in a \$12 loss. However, as the number of trades increases, the orange line, which represents the sum of actual profits, runs closer to the blue line, which represents the expected profit. At 100 trades in this series, the total net profit is \$116 - quite close to the expected profit of \$100.

## Implications

There are four main consequences of the law of large numbers that influence traders in different ways:

1. Strategy testing. You have to test your strategy on as long periods as possible to produce as many test trades as possible for the results to be reliable. What if you come up with a great strategy but test it only 10 trades and get results similar to those above? Or, test a really poor strategy and get a lucky streak of 10 trades? As a trader, you should only trust tests that produced a large number of trades. Unfortunately, it is not always possible, especially when dealing with long-term trading strategies.
2. Deciding between two strategies. When you have two backtested strategies that are both profitable, and one trades much more frequently than the other, it is always better to stick to the former. The calculated expectancy (average profit per trade) might be better with the latter (a strategy with smaller number of trades), but the LLN suggests that we would get more consistent results with the first strategy.